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Theorem pythagtriplem14 16457

Description: Lemma for pythagtrip 16463. Calculate the square of 𝑁. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Hypothesis**
Ref	Expression
pythagtriplem13.1	⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)

**Assertion**
Ref	Expression
pythagtriplem14	⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑁↑2) = ((𝐶 − 𝐴) / 2))

**Proof of Theorem pythagtriplem14**
Step	Hyp	Ref	Expression
1		pythagtriplem13.1	. . 3 ⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)
2	1	oveq1i 7265	. 2 ⊢ (𝑁↑2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2)
3		nncn 11911	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
4		nncn 11911	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5		addcl 10884	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
6	3, 4, 5	syl2anr 596	. . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
7	6	sqrtcld 15077	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
8		subcl 11150	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ)
9	3, 4, 8	syl2anr 596	. . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℂ)
10	9	sqrtcld 15077	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 − 𝐵)) ∈ ℂ)
11	7, 10	subcld 11262	. . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ)
12	11	3adant1 1128	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ)
13	12	3ad2ant1 1131	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ)
14		2cn 11978	. . . . 5 ⊢ 2 ∈ ℂ
15		2ne0 12007	. . . . 5 ⊢ 2 ≠ 0
16		sqdiv 13769	. . . . 5 ⊢ ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)))
17	14, 15, 16	mp3an23 1451	. . . 4 ⊢ (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)))
18	13, 17	syl 17	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)))
19	14	sqvali 13825	. . . . 5 ⊢ (2↑2) = (2 · 2)
20	19	oveq2i 7266	. . . 4 ⊢ ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2))
21	13	sqcld 13790	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) ∈ ℂ)
22		2cnne0 12113	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
23		divdiv1 11616	. . . . . . 7 ⊢ (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) / 2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)))
24	22, 22, 23	mp3an23 1451	. . . . . 6 ⊢ ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) ∈ ℂ → (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) / 2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)))
25	21, 24	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) / 2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)))
26		simp12 1202	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ)
27		simp13 1203	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℕ)
28	26, 27, 7	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
29	26, 27, 10	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℂ)
30		binom2sub 13863	. . . . . . . . . 10 ⊢ (((√‘(𝐶 + 𝐵)) ∈ ℂ ∧ (√‘(𝐶 − 𝐵)) ∈ ℂ) → (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)))
31	28, 29, 30	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)))
32		nnre 11910	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
33		nnre 11910	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
34		readdcl 10885	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ)
35	32, 33, 34	syl2anr 596	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
36	35	3adant1 1128	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
37	36	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℝ)
38	37	recnd 10934	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℂ)
39		resubcl 11215	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ)
40	32, 33, 39	syl2anr 596	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ)
41	40	3adant1 1128	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ)
42	41	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℝ)
43	42	recnd 10934	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℂ)
44	7	3adant1 1128	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
45	10	3adant1 1128	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 − 𝐵)) ∈ ℂ)
46	44, 45	mulcld 10926	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) ∈ ℂ)
47		mulcl 10886	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) ∈ ℂ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
48	14, 46, 47	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
49	48	3ad2ant1 1131	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
50	38, 43, 49	addsubd 11283	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (𝐶 − 𝐵)) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) = (((𝐶 + 𝐵) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)))
51	27	nncnd 11919	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
52		simp11 1201	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℕ)
53	52	nncnd 11919	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℂ)
54		subdi 11338	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · (𝐶 − 𝐴)) = ((2 · 𝐶) − (2 · 𝐴)))
55	14, 51, 53, 54	mp3an2i 1464	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 − 𝐴)) = ((2 · 𝐶) − (2 · 𝐴)))
56		ppncan 11193	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
57	56	3anidm13 1418	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
58		2times 12039	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∈ ℂ → (2 · 𝐶) = (𝐶 + 𝐶))
59	58	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐶) = (𝐶 + 𝐶))
60	57, 59	eqtr4d 2781	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
61	3, 4, 60	syl2anr 596	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
62	61	3adant1 1128	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
63	62	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
64	26	nncnd 11919	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
65		subsq 13854	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
66	51, 64, 65	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
67		oveq1 7262	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
68	67	3ad2ant2 1132	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
69		nncn 11911	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
70	69	sqcld 13790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℂ)
71	70	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℂ)
72	4	sqcld 13790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ ℕ → (𝐵↑2) ∈ ℂ)
73	72	3ad2ant2 1132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℂ)
74	71, 73	pncand 11263	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = (𝐴↑2))
75	74	3ad2ant1 1131	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = (𝐴↑2))
76	68, 75	eqtr3d 2780	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶↑2) − (𝐵↑2)) = (𝐴↑2))
77	66, 76	eqtr3d 2780	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) · (𝐶 − 𝐵)) = (𝐴↑2))
78	77	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶 − 𝐵))) = (√‘(𝐴↑2)))
79	32	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
80	33	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
81		nngt0 11934	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 ∈ ℕ → 0 < 𝐶)
82	81	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
83		nngt0 11934	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
84	83	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
85	79, 80, 82, 84	addgt0d 11480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 + 𝐵))
86		0re 10908	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ
87		ltle 10994	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ ∧ (𝐶 + 𝐵) ∈ ℝ) → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
88	86, 87	mpan 686	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 + 𝐵) ∈ ℝ → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
89	35, 85, 88	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ (𝐶 + 𝐵))
90	89	3adant1 1128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ (𝐶 + 𝐵))
91	90	3ad2ant1 1131	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 + 𝐵))
92		pythagtriplem10 16449	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶 − 𝐵))
93	92	3adant3 1130	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 − 𝐵))
94		ltle 10994	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ (𝐶 − 𝐵) ∈ ℝ) → (0 < (𝐶 − 𝐵) → 0 ≤ (𝐶 − 𝐵)))
95	86, 94	mpan 686	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 − 𝐵) ∈ ℝ → (0 < (𝐶 − 𝐵) → 0 ≤ (𝐶 − 𝐵)))
96	42, 93, 95	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 − 𝐵))
97	37, 91, 42, 96	sqrtmuld 15064	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶 − 𝐵))) = ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))
98	78, 97	eqtr3d 2780	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐴↑2)) = ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))
99		nnre 11910	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
100	99	3ad2ant1 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
101	100	3ad2ant1 1131	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℝ)
102		nnnn0 12170	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
103	102	nn0ge0d 12226	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℕ → 0 ≤ 𝐴)
104	103	3ad2ant1 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ 𝐴)
105	104	3ad2ant1 1131	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ 𝐴)
106	101, 105	sqrtsqd 15059	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐴↑2)) = 𝐴)
107	98, 106	eqtr3d 2780	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) = 𝐴)
108	107	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) = (2 · 𝐴))
109	63, 108	oveq12d 7273	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (𝐶 − 𝐵)) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) = ((2 · 𝐶) − (2 · 𝐴)))
110	55, 109	eqtr4d 2781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 − 𝐴)) = (((𝐶 + 𝐵) + (𝐶 − 𝐵)) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))))
111		resqrtth 14895	. . . . . . . . . . . . 13 ⊢ (((𝐶 + 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 + 𝐵)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
112	37, 91, 111	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
113	112	oveq1d 7270	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) = ((𝐶 + 𝐵) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))))
114		resqrtth 14895	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 − 𝐵)) → ((√‘(𝐶 − 𝐵))↑2) = (𝐶 − 𝐵))
115	42, 96, 114	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 − 𝐵))↑2) = (𝐶 − 𝐵))
116	113, 115	oveq12d 7273	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)) = (((𝐶 + 𝐵) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)))
117	50, 110, 116	3eqtr4rd 2789	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)) = (2 · (𝐶 − 𝐴)))
118	31, 117	eqtrd 2778	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) = (2 · (𝐶 − 𝐴)))
119	118	oveq1d 7270	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) = ((2 · (𝐶 − 𝐴)) / 2))
120		subcl 11150	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐶 − 𝐴) ∈ ℂ)
121	3, 69, 120	syl2anr 596	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐴) ∈ ℂ)
122	121	3adant2 1129	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐴) ∈ ℂ)
123	122	3ad2ant1 1131	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐴) ∈ ℂ)
124		divcan3 11589	. . . . . . . . 9 ⊢ (((𝐶 − 𝐴) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · (𝐶 − 𝐴)) / 2) = (𝐶 − 𝐴))
125	14, 15, 124	mp3an23 1451	. . . . . . . 8 ⊢ ((𝐶 − 𝐴) ∈ ℂ → ((2 · (𝐶 − 𝐴)) / 2) = (𝐶 − 𝐴))
126	123, 125	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 · (𝐶 − 𝐴)) / 2) = (𝐶 − 𝐴))
127	119, 126	eqtrd 2778	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) = (𝐶 − 𝐴))
128	127	oveq1d 7270	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) / 2) = ((𝐶 − 𝐴) / 2))
129	25, 128	eqtr3d 2780	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)) = ((𝐶 − 𝐴) / 2))
130	20, 129	eqtrid 2790	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)) = ((𝐶 − 𝐴) / 2))
131	18, 130	eqtrd 2778	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2) = ((𝐶 − 𝐴) / 2))
132	2, 131	eqtrid 2790	1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑁↑2) = ((𝐶 − 𝐴) / 2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 ↑cexp 13710 √csqrt 14872 ∥ cdvds 15891 gcd cgcd 16129

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875

This theorem is referenced by: pythagtriplem15 16458 pythagtriplem17 16460

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